Question #43653

A spherical blackbody of radius $R$ is maintained at a temperature $T$ such that $k_B T \ll hc R_H$, where $R_H$ is the Rydberg constant. It is surrounded by a thin, optically thick shell of hydrogen gas in its ground state. The shell absorbs all incident photons from the blackbody that have an energy greater than or equal to the ionization energy of hydrogen. Assume that each absorbed ionizing photon causes exactly one ionization event, and the subsequent recombination cascade produces exactly one Lyman-$\alpha$ photon (transition from $n=2$ to $n=1$).

Using Wien's approximation for the spectral emissive power (power emitted per unit area per unit wavelength) $E_\lambda = \frac{2\pi hc^2}{\lambda^5} e^{-hc/\lambda k_B T}$, determine the total power $P_{L\alpha}$ emitted by the hydrogen shell in the Lyman-$\alpha$ line.

(Assume $\int_{x_0}^\infty x^n e^{-x} dx \approx x_0^n e^{-x_0}$ for $x_0 \gg 1$, where $x = \frac{hc}{\lambda k_B T}$ and $x_0$ corresponds to the ionization threshold).